This is an assumption made in Marcus, Number Fields.
$\textbf Z[\omega]$ is UFD for $\omega$ $p-$th root of unity with $3 < p < 23$.
For $p = 2, 3$, one can use Neukrich geometric proof on the distance to the nearest lattice point. It seems that this proof fails for $p=5$ as there are lattice points which is not generated by shifting the fundamental lattice $0,\omega,i\omega,(1+i)\omega$.
Q: Was there a unified approach to deal with this problem for $3 < p < 23$?